Mines mathematics

The grid

A round is a 5 × 5 grid: 25 tiles, of which you choose 1 to 24 to be mines. The rest are safe. Each safe tile you reveal multiplies your stake; one mine ends the round with the stake lost. You can cash out after any safe reveal.

Where the multiplier comes from

Revealing a safe tile is a draw without replacement. With m mines and i tiles already revealed, the chance the next tile is safe is (25 − m − i) / (25 − i). The fair multiplier for surviving n reveals is the inverse of the probability of surviving all of them, and Mines applies its 2% edge once on top:

multiplier(n, m) = 0.98 × Π_(i=0..n−1) (25 − i) / (25 − m − i)

Three mines, two safe reveals: 25/22 × 24/21 = 1.299, times 0.98 = 1.27×. Every cell in the table below is that formula.

Every setting costs the same

Multiply any multiplier by the probability of reaching it and you get 0.98. Ten mines and eight reveals pays 164.72× because the chance of getting there is about 0.6%; one mine and one reveal pays 1.02× because the chance is 96%. The house keeps 2% of expected turnover in both cases. Choosing more mines, or clicking one more tile, changes volatility, never value. That trade-off is the subject of Bankroll and volatility.

Why the mines cannot move

Before your first click the server has already computed the full grid from the provably fair hash: it walks the HMAC-SHA256 output in 8-character chunks, maps each chunk to a tile, and places mines until it has m of them, re-hashing if it runs out of chunks. The seed hash was committed to you before the bet, so the layout is fixed and later verifiable. Details of the hash itself are in Provably Fair, explained.